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Course: Mathematics for Physicists

Department/Abbreviation: OPT/SZZL2

Year: 2021

Guarantee: 'prof. Mgr. Jaromír Fiurášek, Ph.D.'

Annotation: The final exam shall verify knowledge and understanding of mathematical methods employed in physics, that the students learned during their whole study.

Course review:
1. The notion of function, limit of a function, derivative, maxima and minima of a function, power series, antiderivatives, the definite integral. Calculation of areas, volumes, lengths, tangents, centers of gravity, moments of inertia. 2. Functions of several variables, partial derivatives, extrema of functions of several variables, gradient, divergence, curl and their applications, Gauss-Ostrogradsky and Stokes theorems. Basic formulas of analytical geometry, trigonometry, stereometry and plane geometry. 3. Vectors, scalars, tensors, vector spaces, bases, dot product, cross product, coordinate transformations, linear transformations. Basic tensors in physics, tensor of inertia, stress tensor, and metric tensor. 4. Matrices, eigenvalues ??and eigenvectors, characteristic polynomials, orthogonal and unitary matrices, linear operators, spectrum of an operator. Quadratic forms, diagonalization, conics and quadrics. Determinants, solving systems of linear equations. 5. Ordinary differential equations and their systems. Solution of the first order equations by separation of variables, Lagrange method of variation of parameters, Laplace transforms. Linear differential equations with constant coefficients. Solution of inhomogeneous equations. Linear and nonlinear oscillations in physics, coupled oscillators. 6. Algebra of complex numbers, functions of complex variables, holomorphic functions, Cauchy's integral theorem, residue theorem, Fourier series, Fourier transform and its applications. Phasors in physics, conformal mappings in hydrodynamics. 7. Random events, random variable, distribution of discrete and continuous random variables, characteristic function, moments and cumulants. Statistical evaluation of the experimental data. 8. Parciální differential equations, classification of the second order quasilinear PDEs, classification of problems according to the initial and boundary conditions. PDEs in physics: the wave equation, Helmholtz equation, Laplace equation, Maxwell's equations, Euler equation, Navier-Stokes equation, Schroedinger equation, heat equation, the diffusion equation. 9. The wave equation and its solution, d'Alembert's formula, wave reflection, solutions of PDEs by separation of variables, solution by Fourier transformation. 10. Harmonic functions, maximum principle. Numerical methods for solving the partial differential equations, finite difference method, finite element method. Calculation of the gravitational, electric and magnetic fields for given source distributions.